.. _sphx_glr_api_gallery_large_scale:

Large lattices
--------------

Lattices far beyond dense diagonalization: sparse neighbour search,
shift-invert eigensolvers, and the kernel polynomial method, which
turns spectral densities and conductivities of hundreds of thousands of
sites into a few hundred sparse matrix-vector products. In k-space,
fine meshes are diagonalized in batches, optionally across threads.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The band and topology tools of KSpace diagonalize H(\mathbf{k}) at every point of a mesh. Calling the eigensolver once per k-point spends most of its time in Python, not in LAPACK. tbkit builds all the Bloch matrices with one vectorized Bloch sum, then diagonalizes them with stacked LAPACK calls (``numpy.linalg.eigh`` on an array of shape (nk, norb, norb)), in chunks of bounded memory. set_workers spreads the chunks over threads (LAPACK releases the GIL).">

.. only:: html

  .. image:: /api/gallery/large_scale/images/thumb/sphx_glr_plot_batched_k_space_thumb.png
    :alt:

  :doc:`/api/gallery/large_scale/plot_batched_k_space`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Batched and Threaded k-Space Diagonalization: the Haldane Model on 90,000 k-Points</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Diagonalizing N sites costs N^3 operations and N^2 memory: a few thousand sites at most. The kernel polynomial method (Silver and Roder, 1994; reviewed by Weisse, Wellein, Alvermann and Fehske, Rev. Mod. Phys. 78, 275 (2006)) never diagonalizes. It expands the spectral density in Chebyshev polynomials of the rescaled Hamiltonian,">

.. only:: html

  .. image:: /api/gallery/large_scale/images/thumb/sphx_glr_plot_kernel_polynomial_method_thumb.png
    :alt:

  :doc:`/api/gallery/large_scale/plot_kernel_polynomial_method`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">The Kernel Polynomial Method: Graphene With Vacancies, 45,000 Sites</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:

   /api/gallery/large_scale/plot_batched_k_space
   /api/gallery/large_scale/plot_kernel_polynomial_method